DIAGNOSTICS OF MATHEMATICAL PROOF OF THE BEAL CONJECTURE IN MEDICAL PSYCHOLOGY (REMAKE OF PREVIOUS AUTHOR’S ARTICLES CONCERNING FERMAT’S LAST THEOREM)

In the given work diagnostics of mathematical proof of the Beal Conjecture (Generalized Fermat’s Last Theorem) obtained in the earlier author’s works was conducted and truthfulness of the suggested proof was established. Realizing the process of the Bill Conjecture solution, the mathematical structure defining hypothetical equality of the Fermat theorem was determined. Such a structure turned to be one of Pythagorean theorem with whole numbers. With help of Euclid’s geometrical theorem and Fermat’s method of infinite descent one can manage to set that Pythagorean equation in whole numbers representing Fermat’s Last Theorem cannot exist and then the Fermat theorem is true, that is Fermat’s equality in natural numbers does not exist. Thus mental scheme of “demonstratio mirabile”, which Pierre de Fermat mentioned on the margins of Diophantus’s “Arithmetic”, was reconstructed.

On The Diophantine equation (x^n-1)(y^n-1)=z^n-1

Using a variety of techniques, including the hypergeometric method of Thue and Siegel, as well as an assortment of gap principles, M . Bennett proved that the Diophantine equation (x^n-1 ) (y^n-1)=z^n-1 has only the solutions (x;y;z;n)=(-1;4;-5;3) and (4;-1;-5;3) in integers x;y;z and n with |z|>1 and n>2. We aim to prove them in very weak systems using elementary function of arithmetic (EFA), On a new, easy and simple, it's the combination of congruence and infinite descent of Fermat.

Proof of Fermat's Last Theorem By Infinite Descent

Fermat's last theorem, one of the most challenging theories in the history of mathematics, has been conjectured by French lawyer Pierre de Verma in 1637. Since then, it wasconsidered the most difficult and unsolvable mathematical problem. However, more than three centuries later, a first proof was proposed by the British mathematician Andrew Wiles in 1994, relying on 20th-century techniques. Wiles's proof is based on elliptic (oval) curves that were not available at the time when the theory was first proposed. Most mathematicians argued that it was impossible to prove Fermat's theorem according to basic principles of arithmetic, though Harvey Friedman's grand conjecture states that mathematical theorems, including Fermat's Last Theorem, can be solved in very weak systems such as the Elementary Function Arithmetic (EFA). Friedman's grand conjecture states that "every theorem published in the journal, Annals of Mathematics, whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA, which is the weak fragment of Peano Arithmetic based on the usual quantifier free axioms for 0,1,+,x, exp, together with thescheme of induction for all formulas in the language all of whose quantifiers are bounded." *

Fermat's last theorem proved by induction

A proof of Fermat’s last theorem is demonstrated. It is very brief, simple, elementary, and absolutely arithmetical. The necessary premises for the proof are only: the property of identity of the relation of equality, modus tollens, axiom of induction, the proof of Fermat’s last theorem in the case of “n = 3” as well as the premises necessary for the formulation of the theorem itself. It involves a modification of Fermat’s approach of infinite descent. The infinite descent is linked to induction starting from “n = 3” by modus tollens. An inductive series of modus tollens is constructed. The proof of the series by induction is equivalent to Fermat’s last theorem. As far as Fermat had been proved the theorem for “n = 4”, one can suggest that the proof at least for “n ≥ 4” had been accessible to him.

Two criteria in Fermat infinite descent method

A method infinite descent is traditionally used to proof the Fermat’s theorem for the special case of exponent n=4. At each step, the method sequentially generates a new Fermat’s equation with one of the term being smaller than that in the preceding step. After a finite number of steps the term becomes less than one and this is taken as criterion of the insolvency of the original Fermat’s equation. We show that the power of factor 2, in even parameter of Pythagoras’ equation solution used in the proof, decreases by one at each step of the descent. As a result we arrive at an unsolvable equation. This is the second criterion for the descent method. Which of the two criteria is reached first depends on the parameters of the initial Pythagorean solutions chosen for the analysis.